unique adj

1 radically distinctive and without equal; "he is alone in the field of microbiology"; "this theory is altogether alone in its penetration of the problem"; "Bach was unique in his handling of counterpoint"; "craftsmen whose skill is unequaled"; "unparalleled athletic ability"; "a breakdown of law unparalleled in our history" [syn: alone(p), unequaled, unequalled, unparalleled]

2 (followed by `to') applying exclusively to a given category or condition or locality; "a species unique to Australia" [syn: unique(p)]

3 the single one of its kind; "a singular example"; "the unique existing example of Donne's handwriting"; "a unique copy of an ancient manuscript"; "certain types of problems have unique solutions" [syn: singular]

4 highly unusual or rare but not the single instance; "spoke with a unique accent"; "had unique ability in raising funds"; "a frankness unique in literature"; "a unique dining experience"

English

Etymology

From unique.

Pronunciation

• Rhymes: -iːk

Adjective

unique (comparable only in disputed usage)

1. Being only one of its self.
2. (disputed) Of a rare quality.
3. (disputed) Unusual.

Usage notes

• Unique in its undisputed sense is not comparable. In its disputed senses, however, it can be compared as more unique, most unique, and occasionally as uniquer, uniquest.

Derived terms

• uniqueness

Related terms

• unicity

Translations

French

Etymology

From unicus.

Pronunciation

• IPA: /y.nik/
• SAMPA: /y.nik/

Adjective

1. unique
2. only

Related terms

• unicité
• uniquement

In mathematics and logic, the phrase "there is one and only one" is used to indicate that exactly one object with a certain property exists. In mathematical logic, this sort of quantification is known as uniqueness quantification or unique existential quantification.

Uniqueness quantification is denoted with the symbol "∃!". For example, the formal statement

\exists! n \in \mathbb\,(n - 2 = 4)

may be read aloud as "there is exactly one natural number n such that n - 2 = 4".

Reduction to ordinary existential and universal quantification

Uniqueness quantification can be expressed in terms of the existential and universal quantifiers of predicate logic by defining the formula ∃!x P(x) to mean

\exists x\,(P(x) \And \forall y\,(P(y) \to x = y)).

An equivalent definition that has the virtue of separating the notions of existence and uniqueness into two clauses, at the expense of brevity, is

\exists x\,P(x) \And \forall y\, \forall z\,(P(y) \And P(z) \to y = z).

Another equivalent definition with the advantage of brevity is

\exists x\,\forall y\,(x = y \leftrightarrow P(x)).

Generalizations

One generalization of uniqueness quantification is counting quantification. This includes both quantification of the form "exactly k objects exist such that ..." as well as "infinitely many objects exist such that ..." and "only finitely many object exist such that...". The first of these forms is expressible using ordinary quantifiers, but the latter two cannot be expressed in ordinary first-order logic.

See also

• Well-defined

a certain, absolute, almost unheard-of, alone, an, another, any, any one, appropriate, atomic, authentic, avant-garde, azygous, beguiling, bewildering, beyond compare, beyond comparison, breakaway, celibate, characteristic, creative, different, distinctive, distinguished, easily first, either, else, enigmatic, exceptional, exclusive, extraordinary, fabulous, facile princeps, fantastic, fascinating, first and last, firsthand, fresh, idiocratic, idiosyncratic, imaginative, immortal, impair, in character, incomparable, incomprehensible, inconceivable, incredible, individual, indivisible, infrequent, inimitable, integral, intrinsic, invincible, irreducible, lone, marked, marvelous, matchless, miraculous, monadic, monistic, new, not that sort, not the same, not the type, novel, nulli secundus, odd, of a sort, of another sort, of sorts, offbeat, one, one and only, only, only-begotten, original, other, other than, otherwise, outlandish, particular, passing strange, peculiar, peerless, phenomenal, prodigious, proper, puzzling, quintessential, rare, recherche, remarkable, revolutionary, scarce, second to none, seldom met with, seldom seen, sensational, separate, simple, single, singular, slow, sole, solid, solitary, solo, sparse, special, strange, striking, stupendous, sui generis, true to form, unanalyzable, unapproachable, unapproached, unbeatable, uncommon, uncustomary, underived, undivided, undreamed-of, unequaled, unexampled, unexcelled, unexpected, unfamiliar, unfrequent, unheard-of, uniform, unimaginable, unitary, unmatchable, unmatched, unordinary, unpaired, unparagoned, unparalleled, unpeered, unprecedented, unrepeatable, unrepeated, unrivaled, unsurpassable, unsurpassed, unthinkable, unthought-of, unusual, unwonted, whole, without equal, without parallel, wonderful, wondrous